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Theorem fin4en1 8115
Description: Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
fin4en1  |-  ( A 
~~  B  ->  ( A  e. FinIV  ->  B  e. FinIV ) )

Proof of Theorem fin4en1
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ensym 7085 . 2  |-  ( A 
~~  B  ->  B  ~~  A )
2 bren 7046 . . . 4  |-  ( B 
~~  A  <->  E. f 
f : B -1-1-onto-> A )
3 simpr 448 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  ->  x  C.  B )
4 f1of1 5606 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  f : B -1-1-> A )
5 pssss 3378 . . . . . . . . . . . . . 14  |-  ( x 
C.  B  ->  x  C_  B )
6 ssid 3303 . . . . . . . . . . . . . 14  |-  B  C_  B
75, 6jctir 525 . . . . . . . . . . . . 13  |-  ( x 
C.  B  ->  (
x  C_  B  /\  B  C_  B ) )
8 f1imapss 5944 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  ( x  C_  B  /\  B  C_  B ) )  ->  ( (
f " x ) 
C.  ( f " B )  <->  x  C.  B ) )
94, 7, 8syl2an 464 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
x  C.  B )
)
103, 9mpbird 224 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  ( f " B ) )
11 imadmrn 5148 . . . . . . . . . . . . . 14  |-  ( f
" dom  f )  =  ran  f
12 f1odm 5611 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  ->  dom  f  =  B )
1312imaeq2d 5136 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ( f " dom  f )  =  ( f " B
) )
14 dff1o5 5616 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> A  <->  ( f : B -1-1-> A  /\  ran  f  =  A ) )
1514simprbi 451 . . . . . . . . . . . . . 14  |-  ( f : B -1-1-onto-> A  ->  ran  f  =  A )
1611, 13, 153eqtr3a 2436 . . . . . . . . . . . . 13  |-  ( f : B -1-1-onto-> A  ->  ( f " B )  =  A )
1716adantr 452 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " B
)  =  A )
1817psseq2d 3376 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( ( f "
x )  C.  (
f " B )  <-> 
( f " x
)  C.  A )
)
1910, 18mpbid 202 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  C.  A )
2019adantrr 698 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  C.  A
)
21 vex 2895 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2221f1imaen 7098 . . . . . . . . . . . . 13  |-  ( ( f : B -1-1-> A  /\  x  C_  B )  ->  ( f "
x )  ~~  x
)
234, 5, 22syl2an 464 . . . . . . . . . . . 12  |-  ( ( f : B -1-1-onto-> A  /\  x  C.  B )  -> 
( f " x
)  ~~  x )
2423adantrr 698 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  x
)
25 simprr 734 . . . . . . . . . . 11  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  x  ~~  B
)
26 entr 7088 . . . . . . . . . . 11  |-  ( ( ( f " x
)  ~~  x  /\  x  ~~  B )  -> 
( f " x
)  ~~  B )
2724, 25, 26syl2anc 643 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  B
)
28 vex 2895 . . . . . . . . . . . 12  |-  f  e. 
_V
29 f1oen3g 7052 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : B -1-1-onto-> A )  ->  B  ~~  A )
3028, 29mpan 652 . . . . . . . . . . 11  |-  ( f : B -1-1-onto-> A  ->  B  ~~  A )
3130adantr 452 . . . . . . . . . 10  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  B  ~~  A
)
32 entr 7088 . . . . . . . . . 10  |-  ( ( ( f " x
)  ~~  B  /\  B  ~~  A )  -> 
( f " x
)  ~~  A )
3327, 31, 32syl2anc 643 . . . . . . . . 9  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  ( f "
x )  ~~  A
)
34 fin4i 8104 . . . . . . . . 9  |-  ( ( ( f " x
)  C.  A  /\  ( f " x
)  ~~  A )  ->  -.  A  e. FinIV )
3520, 33, 34syl2anc 643 . . . . . . . 8  |-  ( ( f : B -1-1-onto-> A  /\  ( x  C.  B  /\  x  ~~  B ) )  ->  -.  A  e. FinIV )
3635ex 424 . . . . . . 7  |-  ( f : B -1-1-onto-> A  ->  ( (
x  C.  B  /\  x  ~~  B )  ->  -.  A  e. FinIV ) )
3736exlimdv 1643 . . . . . 6  |-  ( f : B -1-1-onto-> A  ->  ( E. x ( x  C.  B  /\  x  ~~  B
)  ->  -.  A  e. FinIV
) )
3837con2d 109 . . . . 5  |-  ( f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
3938exlimiv 1641 . . . 4  |-  ( E. f  f : B -1-1-onto-> A  ->  ( A  e. FinIV  ->  -.  E. x ( x  C.  B  /\  x  ~~  B
) ) )
402, 39sylbi 188 . . 3  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  -.  E. x
( x  C.  B  /\  x  ~~  B ) ) )
41 relen 7043 . . . . 5  |-  Rel  ~~
4241brrelexi 4851 . . . 4  |-  ( B 
~~  A  ->  B  e.  _V )
43 isfin4 8103 . . . 4  |-  ( B  e.  _V  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
4442, 43syl 16 . . 3  |-  ( B 
~~  A  ->  ( B  e. FinIV 
<->  -.  E. x ( x  C.  B  /\  x  ~~  B ) ) )
4540, 44sylibrd 226 . 2  |-  ( B 
~~  A  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
461, 45syl 16 1  |-  ( A 
~~  B  ->  ( A  e. FinIV  ->  B  e. FinIV ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2892    C_ wss 3256    C. wpss 3257   class class class wbr 4146   dom cdm 4811   ran crn 4812   "cima 4814   -1-1->wf1 5384   -1-1-onto->wf1o 5386    ~~ cen 7035  FinIVcfin4 8086
This theorem is referenced by:  domfin4  8117  isfin4-3  8121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-er 6834  df-en 7039  df-fin4 8093
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